Lesson 2

Example 1: Formulating a Quadratic Equation from a Graph

 

In Try This 3 you sketched a graph from an equation. Now you will do the opposite—you will formulate a quadratic equation from a graph.

 

Question

 

What quadratic function describes the parabola on the grid below?

 

 

Solution

 

Step 1: Find the variables p and q. Since the vertex coordinates are (p, q), you can read the coordinates of the vertex of the parabola from the graph as (−5, −4).

 

p = −5
q = −4

 

Step 2: Substitute p and q into the quadratic function equation.

 

 

Step 3: Find the value of a. The graph opens upward, so you know the value of a is positive. You can pick another point on the graph and substitute its coordinates into the function to solve for a. An easy point is the y-intercept, which is at (0, 1).

 

 

Step 4: Substitute in the variables. Put the found values of a, p, and q into the equation for a quadratic function.

 

y = 0.2(x + 5)2 − 4

 

Step 5: Verify the equation. Verify the equation by substituting the coordinates of a second point into the equation and see if the equation is true. For example, a second point shown on the parabola is (−1, −0.8). The left side equals the right side, so the equation has been verified.

Left Side	Right Side
(4)2y (4)2−0.8 (4)2−0.8 (4)2−0.8 −0.8 −0.8	0.2(x + 5)2 − 4 0.2((−1) + 5)2 − 4 0.2(4)2 − 4 0.2(16) − 4 3.2 − 4 −0.8

 

Self-Check 3

1. What quadratic function describes the parabola on the grid shown?
   
   
    
   
   Answer
   

    

2. What quadratic function describes a parabola with the following characteristics?
   
   

   • The parabola opens downward.
   • The vertex of the parabola is at (4, 6).
   • There are two x-intercepts. One is at (6, 0) and the other is at (2, 0).
   • The y-intercept is at (0, −18).
     
     

   Answer